The easiest way to achieve this is to use a map package which will contain a map document (.mxd) and the data referenced by the layers it contains (a file geodatabase), packaged into one convenient, portable file (which is really just an archive you can unpack with archiving software such as 7Zip).
Map packages can be used for easy sharing of maps between colleagues
in a work group, across departments in an organization, or with any
other ArcGIS users via ArcGIS Online. Map packages have other uses,
too, such as the ability to create an archive of a particular map that
contains a snapshot of the current state of the data used in the map.
Unpack the package and now you are ready to send the geodatabase with the data and the map document with paths referring to this geodatabase. An output .mxd map document will contain relative paths to the exported geodatabase.
You can also use Python and arcpy.mapping module and geoprocessing tools (with/without ModelBuilder) to automate the process of data source paths changes to produce a new map document and a geodatabase containing the data needed.